Problem :
I am looking to construct maximal latin rectangles $r\times n$ that do not contain smaller latin rectangles.
More formally, for any two rows $\{p_1,...,p_n\}$ and $\{q_1,...,q_n\}$, I want to have :
For all non-empty $C\subset \{1,...,n\}$, $\bigcup_{i\in C}\{p_i\}\neq \bigcup_{i\in C}\{q_i\}$
Example :
\begin{array}{c c c c c c} 1 & 2 & 3 & 4 & 5 & 6\\ 6 & 1 & 2 & 3 & 4 & 5 \end{array}
works whereas
\begin{array}{c c c c c c} 1 & 2 & 3 & 4 & 5 & 6\\ 6 & 1 & 2 & 3 & 4 & 5\\ 5 & 6 & 1 & 2 & 3 & 4 \end{array}
does not work as both the first and third rows have $\{1,3,5\}$ on columns 1, 3 and 5 and $\{2,4,6\}$ on columns 2, 4 and 6.
Questions :
- Is there a name for these sorts of things so that I can look them up myself ? Some kind of generalization of derangements maybe ?
- Is there any known results on the maximal number of rows these rectangles can have for a given n ?
- In the case the answer to the previous question is positive, is there any algorithm for constructing them ?
Thanks in advance,
Authary